particles coincide with the streamlines of the flow, that are isolines of the stream function. In 2-D, streamlines are concentric closed curves that cross only at stagnation points. Thus, a spot of dyed fluid to be mixed can only explore the region bounded by the most external and internal streamline, on which it is lying at the initial time. Regarding practical applications, this configuration is not very satisfying. For 2-D unstationary (time-dependent) flows, instantaneous closed streamlines and Lagrangian trajectories do not coincide any more. Hence, Lagrangian trajectories explore a larger volume of the volume, resulting in better mixing. Chaotic advection is observed for most 2-D unstationary flows. A famous example is the blinking vortex flow introduced by Aref, where two fixed rod-like agitators are alternately rotated inside the fluid. Switching periodically the active (rotating) agitator introduces a time-dependency in the flow, which results in chaotic advection. Lagrangian trajectories can therefore escape from closed streamlines, and visit a large fraction of the fluid domain. Shear A flow promotes mixing by separating neighboring fluid particles. This separation occurs because of velocity gradients, a phenomenon called shearing. Let and be two neighboring fluid particles, separated by at time t. When the particles are advected by a flow , at time the approximate separation between the particles can be found through Taylor expansion : hence and The rate of growth of the separation is therefore given by the gradient of the velocity field in the direction of the separation. The plane shear flow is a simple example of large-scale stationary flow that deforms fluid elements because of a uniform shear. Characterization of chaotic advection Lyapunov exponents If the flow is chaotic, then small initial errors, , in a trajectory will diverge exponentially. We are interested in calculating the stability—i.e., how fast do nearby trajectories diverge? The Jacobi matrix of the velocity field, , provides information about the local rate of divergence of nearby trajectories or the local rate of stretching of Lagrangian space. We define the matrix H such that: where I is the identity matrix. It follows that: The finite-time Lyapunov exponents are defined as the time average of the logarithms of the lengths of the principal components of the vector H over a time t: where is the ith Lyapunov exponent of the system, while is the ith principal component of the matrix H. If we start with a set of orthonormal